Finite Group Databasebeta

← #41014402

Subgroups — #41014402

Order 27008.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16809950C211yesmaximal, characteristic, complemented
2C2×#20269957C211yesmaximal, characteristic, complemented
3C2×#20269948C211yesmaximal, characteristic
4C2×#20269902C211yesmaximal, characteristic, complemented
5C2×(C2×D211)⋊C8C211yesmaximal, characteristic, complemented
6#27212908C211yesmaximal, characteristic, complemented
7C2×#102×C211C211yesmaximal, characteristic, complemented
8C22×Dic422C2211yescharacteristic
9C22×C22⋊D211—21no—
10C22×C4×D211C2211yescharacteristic
11C2×#16809950—22no—
12C2×#16809950—22no—
13C2×#16809950—22no—
14C22×C211⋊C8C2211yescharacteristic
15#20269957—22no—
16#20269957—22no—
17#20269948—22no—
18#20269948—22no—
19C2×C4⋊Dic211—21no—
20C2×C4⋊Dic211C2211yescharacteristic
21C2×C22⋊Dic211—21no—
22C2×(C2×D211)⋊C4—21no—
23#20269902—22no—
24(C2×D211)⋊C8—22no—
25C22×D4×C211C2211yescharacteristic
26C22×C8×C211C2211yescharacteristic
27C2×C4⋊C4×C211C2211yescharacteristic
28#102×C211—22no—
29C23×D211D411yescharacteristic
30C2×Dic422—21no—
31C2×Dic422—21no—
32C2×Dic422—21no—
33C2×Dic422—21no—
34C2×Dic422—21no—
35C2×Dic422D412yescomplemented
36C2×C22⋊D211—24no—
37C2×C22⋊D211—42no—
38C2×C22⋊D211—42no—
39C22×Dic211—21no—
40C22×Dic211—21no—
41C22×Dic211D411yescharacteristic
42C2×C4×D211—21no—
43C2×C4×D211—21no—
44C2×C4×D211—21no—
45C2×C4×D211—21no—
46C2×C4×D211—21no—
47C2×C4×D211—21no—
48#16809950—28no—
49C2×C211⋊C8—21no—
50C2×C211⋊C8—21no—
51C2×C211⋊C8D412yescomplemented
52C4⋊Dic211—24no—
53C4⋊Dic211—21no—
54C4⋊Dic211D412yes—
55C22⋊Dic211—24no—
56(C2×D211)⋊C4—24no—
57C24×C211—21no—
58C2×D4×C211—21no—
59C2×D4×C211—21no—
60C2×D4×C211—21no—
61C2×D4×C211—21no—
62C2×D4×C211—21no—
63C2×D4×C211D412yes—
64C22×C4×C211—21no—
65C22×C4×C211C2311yescharacteristic
66C2×C8×C211—21no—
67C2×C8×C211—21no—
68C2×C8×C211—21no—
69C4⋊C4×C211—21no—
70C4⋊C4×C211D412yescomplemented
71C22×D211—21no—
72C22×D211—21no—
73C22×D211—21no—
74C22×D211—21no—
75C22×D211—21no—
76C22×D211—21no—
77C22×D211SD1612yescomplemented
78Dic422—22no—
79Dic422—22no—
80Dic422—41no—
81Dic422—41no—
82C22⋊D211—48no—
83C2×Dic211—22no—
84C2×Dic211—21no—
85C2×Dic211—22no—
86C2×Dic211—21no—
87C2×Dic211—21no—
88C2×Dic211—41no—
89C2×Dic211—41no—
90C2×Dic211—41no—
91C2×Dic211—41no—
92C4×D211—41no—
93C4×D211—41no—
94C4×D211—41no—
95C4×D211—41no—
96C211⋊C8—22no—
97C23×C211—22no—
98C23×C211—41no—
99C23×C211—41no—
100C23×C211—41no—
101C23×C211—41no—
102C23×C211—41no—
103C23×C211—41no—
104C23×C211C2×D411yescharacteristic
105D4×C211—22no—
106D4×C211—22no—
107D4×C211—41no—
108D4×C211—41no—
109C2×C4×C211—22no—
110C2×C4×C211—41no—
111C2×C4×C211—41no—
112C2×C4×C211C2×D411yescharacteristic
113C2×C4×C211C2×D411yescharacteristic
114C2×C4×C211C2×D411yescharacteristic
115C2×C4×C211C2×D411yescharacteristic
116C2×C4×C211C2×D411yescharacteristic
117C2×C4×C211#4111yescharacteristic
118C8×C211—41no—
119C22×C211—42no—
120C22×C211—42no—
121C22×C211—42no—
122C22×C211—81no—
123C22×C211—81no—
124C22×C211—81no—
125C22×C211—81no—
126C22×C211#12011yescharacteristic
127C22×C211#12011yescharacteristic
128C22×C211#12111yescharacteristic
129C22×C211C2×SD1611yescharacteristic
130C22×C211#13511yescharacteristic
131C22×C211AGL(1,ℤ/8)11yescharacteristic
132C22×C211#13711yescharacteristic
133C2×D211—22no—
134C2×D211—22no—
135C2×D211—22no—
136C2×D211—41no—
137C2×D211—41no—
138C2×D211—41no—
139C2×D211—41no—
140C4×C211—42no—
141C4×C211#12011yescharacteristic
142C4×C211#12111yescharacteristic
143C4×C211#12111yescharacteristic
144C4×C211#12711yescharacteristic
145Dic211—41no—
146Dic211—42no—
147Dic211—42no—
148C2×C211—82no—
149C2×C211#39211yescharacteristic
150C2×C211#44811yescharacteristic
151C2×C211#44911yescharacteristic
152C2×C211#45011yescharacteristic
153C2×C211#45211yescharacteristic
154C2×C211#46511yescharacteristic
155C2×C211#46811yescharacteristic
156D211—42no—
157C211#200011yescharacteristic, complemented, Sylow
158#2000—2111nomaximal, Sylow
159C22×SD16—2111no—
160C2×C22⋊Q8—2111no—
161C2×#121—2111no—
162C2×#103—2111no—
163C2×#102D21111yescharacteristic, complemented
164C2×C22⋊C8—2111no—
165#340—2111no—
166C22×Q8—2111no—
167C22×D4—4221no—
168C22×D4C2×D21111yescharacteristic
169C23×C4—2111no—
170C2×SD16—4222no—
171C2×SD16—4222no—
172C2×SD16—4222no—
173C22×C8—2111no—
174C22×C8C2×D21111yescharacteristic
175C22⋊Q8—4222no—
176C22⋊Q8—4222no—
177#121—4222no—
178#121—4222no—
179C2×C4⋊C4—4221no—
180C2×C4⋊C4—2111no—
181C2×C4⋊C4C2×D21111yescharacteristic
182C2×C22⋊C4—4221no—
183C2×C22⋊C4—4221no—
184#103—4222no—
185#102—22no—
186C22⋊C8—4222no—
187C24—21no—
188C24—2111no—
189C2×Q8—4221no—
190C2×Q8—4221no—
191C2×Q8—4221no—
192C2×Q8—4221no—
193C2×Q8—4221no—
194C2×Q8—2112no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—8442no—
201C2×D4—8442no—
202C2×D4—4224no—
203C2×D4D84412yes—
204C22×C4—21no—
205C22×C4—4221no—
206C22×C4—4221no—
207C22×C4—4221no—
208C22×C4—4221no—
209C22×C4—4221no—
210C22×C4—4221no—
211C22×C4—4221no—
212C22×C4—4221no—
213C22×C4—2111no—
214C22×C4C22×D21111yescharacteristic
215SD16—4228no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—4221no—
220C2×C8—4221no—
221C2×C8—2112no—
222C4⋊C4—21no—
223C4⋊C4—4224no—
224C4⋊C4—4221no—
225C4⋊C4—2112no—
226C4⋊C4C22⋊D21112yescomplemented
227C22⋊C4—4224no—
228C22⋊C4—4224no—
229C23—22no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—4221no—
237C23—4221no—
238C23—4221no—
239C23—4221no—
240C23—4221no—
241C23—4221no—
242C23—2112no—
243C23D4×D21111yescharacteristic
244Q8—8441no—
245Q8—8441no—
246Q8—4222no—
247Q8—4222no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—8448no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—8441no—
257C2×C4—8441no—
258C2×C4—8441no—
259C2×C4—8441no—
260C2×C4—8441no—
261C2×C4—8441no—
262C2×C4—8441no—
263C2×C4—8441no—
264C2×C4—4222no—
265C2×C4—4221no—
266C2×C4—4221no—
267C2×C4—4222no—
268C2×C4—4221no—
269C2×C4D4×D21111yescharacteristic
270C2×C4D4×D21111yescharacteristic
271C2×C4D4×D21111yescharacteristic
272C2×C4C2×C22⋊D21111yescharacteristic
273C2×C4C2×D84411yescharacteristic
274C2×C4#1680996211yescharacteristic
275C8—41no—
276C8—4222no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22—8441no—
285C22—8441no—
286C22—8441no—
287C22—8441no—
288C22—4222no—
289C22—4222no—
290C22—4222no—
291C22#2026994511yescharacteristic
292C22#2026994711yescharacteristic
293C22#2026995011yescharacteristic
294C22SD16×D21111yescharacteristic
295C22#2026999611yescharacteristic
296C22#2027000111yescharacteristic
297C22#2027000611yescharacteristic
298C4—42no—
299C4—8442no—
300C4—8441no—
301C4—8442no—
302C4#2026994711yescharacteristic
303C4#2026995011yescharacteristic
304C4#2026995111yescharacteristic
305C4#2026997311yescharacteristic
306C2—82no—
307C2—8442no—
308C2#2721305511yescharacteristic
309C2#2721328311yescharacteristic
310C2#2721328811yescharacteristic
311C2#2721329811yescharacteristic
312C2#2721330911yescharacteristic
313C2#2721337411yescharacteristic
314C2#2721339111yescharacteristic
subgroupG/H
centerC23D4×D211
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
FittingC2×#102×C211C2
socleC23×C211C2×D4
prime radicalC2×#20269948C2
hypercenterC2×#102D211
hypocenterC211#2000
LCSG ▷ C22×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#16809950C2semidirect
2. C2×#20269957C2semidirect
4. C2×#20269902C2semidirect
5. C2×(C2×D211)⋊C8C2semidirect
6. #27212908C2semidirect
7. C2×#102×C211C2semidirect
35. C2×Dic422D4semidirect
51. C2×C211⋊C8D4semidirect
70. C4⋊C4×C211D4semidirect
77. C22×D211SD16semidirect
157. C211#2000semidirect
163. C2×#102D211semidirect
226. C4⋊C4C22⋊D211semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#168099502
C2×#202699572
C2×#202699482
C2×#202699022
C2×(C2×D211)⋊C82
#272129082
C2×#102×C2112
#2000211